The fifth edition of "Numerical Methods for Engineers" continues its tradition of excellence. Instructors love this text because it is a comprehensive text that is easy to teach from. Students love it because it is written for them--with great pedagogy and clear explanations and examples throughout. The text features a broad array of applications, including all engineering disciplines. The revision retains the successful pedagogy of the prior editions. Chapra and Canale's unique approach opens each part of the text with sections called Motivation, Mathematical Background, and Orientation, preparing the student for what is to come in a motivating and engaging manner. Each part closes with an Epilogue containing sections called Trade-Offs, Important Relationships and Formulas, and Advanced Methods and Additional References. Much more than a summary, the Epilogue deepens understanding of what has been learned and provides a peek into more advanced methods. Approximately 80% of the end-of-chapter problems are revised or new to this edition. The expanded breadth of engineering disciplines covered is especially evident in the problems, which now cover such areas as biotechnology and biomedical engineering. Users will find use of software packages, specifically MATLAB and Excel with VBA. This includes material on developing MATLAB m-files and VBA macros.

Numerical methods for engineers

Summary of low speed airfoil data

Iced-airfoil aerodynamics

Aircraft icing: an ongoing threat to aviation safety

This paper describes a series of experiments on the SD7003 airfoil, conducted in three different facilities: a low-turbulence wind tunnel, a water tunnel, and a tow tank. An attempt was made to achieve commonality of model geometry, Reynolds number and test conditions, in order to directly compare the experimental results. The SD7003 airfoil was chosen because of its robust, thin laminar separation bubble, which challenges the capability of current Particle Image Velocimetry (PIV) methods to accurately resolve the near-wall flowfield. Chordwise locations of separation, reattachment, and transition were compared. Since the time-averaged bubble geometry is a strong function of the flowfield ambient turbulence level, comparison of bubble geometry gives some assessment of facility flow quality, and hence of facility suitability for low Reynolds number testing in general.

Comparison of Laminar Separation Bubble Measurements on a Low Reynolds Number Airfoil in Three Facilities

Recent investigations of two-dimensional airfoil stalling characteristics have revealed low-frequency and highly unsteady e owin somecases and large-scalethree-dimensionalstructuresin other cases.Thelatterwerereferred to as “ stall cells” and can form on two-dimensional cone gurations where the ends of the airfoil model are e ush with tunnelsidewallsorend plates.Thispaperpresentsresultsofdetailedinvestigationsofthestalling characteristicsof several airfoils that exhibited both low-frequency unsteadiness and large-scale three-dimensional structures. The airfoils were wind-tunnel tested in a two-dimensional cone guration. The primary measurements were spanwise wakevelocityand mini-tufte owvisualization. Theresultsshowedthatairfoilswithtrailing-edgeseparationsatand above maximum lift (static stall )exhibited stall-cell patterns. Conversely, airfoils that had leading-edge separation bubblesthatgrewinsizeastheangleofattackwasincreasedintostalldevelopedthelow-frequency,highlyunsteady e ow. This unsteadiness was found to be essentially two dimensional. Therefore, the development of either of these phenomenaappearstobedeterminedbythecharacteristicsoftheboundary-layerseparationleadinguptothestall.

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Spanwise Variation in the Unsteady Stalling Flowfields of Two-Dimensional Airfoil Models

Lift and drag measurements were taken on 34 airfoils at low Reynolds numbers in an attempt to develop a consistent database for use in design studies that require accurate low Reynolds number airfoil data. Prom these data emerged several interesting results related to the behavior of the laminar separation bubbles and their effect on the lift characteristics. A plateau hi the lift curve of symmetrical airfoils hi the vicinity of an angle of attack of 0 deg was found to be common in the Reynolds number range of 40,000 to 100,000. Through the use of zig-zag type boundary-layer trips, this nonlinearity can be reduced owing to a reduction in the size of the laminar separation bubbles. The influence of laminar separation bubbles was also found to dominate the performance of several high-lift airfoils in the Reynolds number range of 80,000 to 150,000. In particular, hysteresis loops in the lift curve were present, and these are related to the size of the laminar separation bubble as deduced from drag data. The data reveals that some airfoils exhibit both counterclockwise aoid clockwise hysteresis loops for a given Reynolds number. Moreover, depending on the airfoil, either type of loop can occur first. Introduction A wide variety of small unmanned aerial vehicles (UAVs) operate in the low Reynolds number regime in which the airfoil aerodynamics play a key role in aircraft performance and handling. In this regime, natural boundary-layer transition takes place through a laminar separation bubble that forms as the laminar boundary layer first separates, then becomes unstable, makes a transition to turbulent flow and reattaches to the airfoil to form a laminar separation bubble.' This bubble often results in a notable airfoil-performance degradation that is characterized by undesirable high drag, nonlinearCopyright © 1996 by Michael S. Selig, James J. Guglielmo, Andy P. Broeren and Philippe Giguere. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. 'Assistant Professor. Member AIAA. fGraduate Research Assistant. Student Member AIAA. ^Graduate Research Assistant. ities in the lift-curve characteristics and sometimes static hysteresis in the section lift, drag and moment data.' In an effort to understand the flow phenomena at low Reynolds numbers, there have been numerous theoretical and experimental investigations that have resulted in three conferences" and a special AGARD publication. Because of the difficulty in modeling the laminar separation bubble, computational efforts, while vital for use in design, have not been entirely reliable hi predicting these complex flows. Nonetheless, considerable progress has been made in recent years." Despite the high level of interest in this area, few systematic experiments have been performed to document the performance of a wide selection of airfoils for use in conceptual and detailed design studies. This has been particularly problematic because comparisons of data between different wind tunnels facilities regularly show discrepencies owing to the documented difficulties in measuring low Reynolds number airfoil performance. Thus, the mixture of different data sets is not ideally suited for the purposes of examining performance trade-offs involving different airfoils. A key objective of the present research was to produce a large and consistent low Reynolds number airfoil performance database for use in the design of small UAVs. Specifically, the experiments involved measuring the lift and drag characteristics of many airfoils over the nominal Reynolds number range of 60,000 to 300,000. The collection of airfoils tested is depicted in Fig. 1. The more limited objective of this paper is to highlight and explain two interesting features observed in the lift characteristics of some of the airfoil tested. In particular, this paper presents and discusses (1) the nonlinear lift characteristics of two symmetrical airfoils (NACA 0009 and SD8020) as well as improvements made through the use of boundary-layer trips, and (2) the hysteresis in the lift characteristics of six high-lift airfoils (FX 63-137, S1210, modified FX 74CL5-140, CH 10-48-13, M06-13-128 and S1223). As will be described, both features, the nonlinearity and the hysteresis of the lift curves, can be linked to the behavior of the laminar separation bubbles. Anti-Turbulence Screens Diffuser Silencer N. / Frequency Controller / Fan \

Experiments on airfoils at low reynolds numbers

Due to numerous engine power-loss events associated with high-altitude convective weather, ice accretion within an engine due to ice-crystal ingestion is being investigated. The National Aeronautics and Space Administration (NASA) and the National Research Council (NRC) of Canada are starting to examine the physical mechanisms of ice accretion on surfaces exposed to ice-crystal and mixed-phase conditions. In November 2010, two weeks of testing occurred at the NRC Research Altitude Facility utilizing a single wedge-type airfoil designed to facilitate fundamental studies while retaining critical features of a compressor stator blade or guide vane. The airfoil was placed in the NRC cascade wind tunnel for both aerodynamic and icing tests. Aerodynamic testing showed excellent agreement compared with CFD data on the icing pressure surface and allowed calculation of heat transfer coefficients at various airfoil locations. Icing tests were performed at Mach numbers of 0.2 to 0.3, total pressures from 93 to 45 kPa, and total temperatures from 5 to 15 C. Ice and liquid water contents ranged up to 20 and 3 grams per cubic meter, respectively. The ice appeared well adhered to the surface in the lowest pressure tests (45 kPa) and, in a particular case, showed continuous leading-edge ice growth to a thickness greater than 15 millimeters in 3 minutes. Such widespread deposits were not observed in the highest pressure tests, where the accretions were limited to a small area around the leading edge. The suction surface was typically ice-free in the tests at high pressure, but not at low pressure. The icing behavior at high and low pressure appeared to be correlated with the wet-bulb temperature, which was estimated to be above 0 C in tests at 93 kPa and below 0 C in tests at lower pressure, the latter enhanced by more evaporative cooling of water. The authors believe that the large ice accretions observed in the low pressure tests would undoubtedly cause the aerodynamic performance of a compressor component such as a stator blade to degrade significantly, and could damage downstream components if shed.

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Fundamental Ice Crystal Accretion Physics Studies

Aerodynamic wind tunnel tests were conducted to study the effects of various ice accretions on the aerodynamic performance of a 36-inch chord, two-dimensional business jet airfoil. Eight different ice shape configurations were tested. Four were castings made from molds of ice shapes accreted in an icing wind tunnel. Two were made using computationally smoothed tracings of two of the ice shapes accreted in the icing tunnel. These smoothed profiles were then extended in the spanwise direction to form a two-dimensional ice shape. The final two configurations were formed by applying grit to the smoothed ice shapes. The ice shapes resulted in as much as 48% reduction in maximum lift coefficient from that of the clean airfoil. Large increases in drag and changes in pitching moment were also observed. The castings and their corresponding smoothed counterparts yielded similar results. Little change in performance was observed with the addition of grit to the smoothed ice shapes. Changes in the Reynolds number (from 3 x 10(exp 6) to 10.5 x 10(exp 6) and Mach number (from 0.12 to 0.28) did not significantly affect the iced-airfoil performance coefficients.

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Andy P. Broeren

Papers

Iced-airfoil aerodynamics

Spanwise Variation in the Unsteady Stalling Flowfields of Two-Dimensional Airfoil Models

Experiments on airfoils at low reynolds numbers

Fundamental Ice Crystal Accretion Physics Studies

A Wind Tunnel Study of Icing Effects on a Business Jet Airfoil